<<<Problem 1>>>

Problem1

The purpose of this exercise is to visualize the Ewald sphere construction
and the inverse relationship between the size of the reciprocal and direct
cells, and to see the effect of moving the crystal on the resulting diffraction
pattern. The effect of changes in wavelength on the reciprocal lattice
is also illustrated. What you are looking at is a side view of a diffraction
experiment, with the X-rays coming in from the left and striking a "crystal"
at the center of the sphere. The X-rays are diffracted, resulting in a
pattern that appears on the surface of a detector (or piece of film) on
the right.

1. Use the pull-down menu labeled "Edit Unit
Cell..." under"Options"
to set the unit cell parameters to a fairly small cell, 10
x15 x 20Angstroms. Click on the button
labeled "OK", after the cell dimensions are set. Notice that
the reciprocal lattice is now relatively large. When the slide bars to
the right are moved, notice that as a reciprocal lattice point touches
Ewald's sphere the color of the lattice point changes to red and a diffracted
ray strikes the detector, producing a "spot" on the detector.
Notice that the spacings between spots on the detector is also large, corresponding
to the large separation between lattice points on the reciprocal lattice.

2. Now change the unit cell to relatively large values, say 50
x 60 x70 Angstroms. (Be patient, it takes a while to generate a
large number of points). Notice that the size of the reciprocal lattice,
and the separation between spots on the detector, has shrunk considerably.
Note also that the number of spots appearing on the detector has increased.
This is a consequence of the smaller reciprocal cell volume. Rotate the
crystal with the slide bars. As the crystal is rotated, rings of spots
can be seen to come in and out of view. These circles arise from the satisfaction
of the von Laue conditions, and represent the intersection of planes in
reciprocal space with the Ewald sphere. The diffracted rays around a circle
lie on the surface of a cone which is often referred to as a Laue cone.

3. Now change the unit cell dimensions to 10 x
10 x 10 Angstroms, change the wavelength to 2
Angstroms, and change the resolution to 2 Angstroms. Notice now
some points are fairly far from the origin of the reciprocal lattice. You
can imagine that at even higher resolution (that is, a reciprocal lattice
with even greater extent) that some lattice points could never be brought
onto the Ewald sphere. This illustrates the concept of the limiting sphere.
For a given wavelength, only those lattice points at resolutions greater
that 1/2 of the wavelength can ever be observed.

<<<Problem 2>>>

Problem 2

This exercise shows how the observation of Laue circles can be used
to precisely align a crystal along a particular axis. Although real crystals
can often be aligned approximately by optical examination of the crystal
faces, examination in reciprocal space is the best way to achieve an accurate
alignment.

1. Set the resolution back to 7 Angstroms
and then set the unit cell to about 50 x 60 x 70
Angstroms with a beta angle of 100
degrees. Switch to a simple view of just the diffraction pattern
by selecting "View Detector" under
the "View" option. Set the axes
labeled "Large" and "Phi"
to zero. You can use the arrow keys for fine adjustments. Now rotate the
crystal around the "Small" axis by putting the pointer on the
slide bar and moving it up and down. Note that the pattern rotates, but
that the particular spots that are shown do not change. This is a consequence
of the circular symmetry of Ewald's sphere about the incident beam and
the fact the the "Small" slider
rotates the crystal about the incident X-ray beam under these conditions.

2. Now you will orient the crystal so that one of its axes is along
the X-ray beam by looking only at the diffraction pattern. use the slide
bar labeled "Large" to align the
crystal so that its 010 ( or y) axis is along the X-ray beam. (This is
a view that will show a non-90 degree angle in the reciprocal lattice.
Remember that beta is the angle between the
a and c axes and is set to 100 degrees at
present. In case you can't find it by wandering around in reciprocal space,
the answer is at about +90 degrees around the axis labeled "Large"
with "Small" and "Phi" set to zero).

<<<Problem 3>>>

Problem3

The mosaic spread of the crystal (and the degree of monochromaticity
of the incident radiation) determine how close a reciprocal lattice point
must be to Ewald's sphere for a "reflection" to appear on the
detector. The wavelength of the incident radiation also effects the size
of the reciprocal lattice.*

1. Use the "Edit Mosaic Spread..."
option under "Options" to observe
the effect of increasing mosaicity on the resulting diffraction pattern.
Note that more spots appear on the detector if the mosaicity, or angular
spread of each reciprocal lattice spot, is increased.

2. Now note the effect of a change in wavelength on the pattern by selecting
the "Edit wavelength" options under
"Options" and clicking on the "Apply" button. (You
should click on the gray bar of the wavelength box and drag it out of the
way, so you can see the diffraction pattern as you change the wavelength.)
Notice that the spots become closer together as the wavelength is decreased
and vice versa. Note also that the particular spots that appear on the
detector also change, corresponding to the different relative curvature
of Ewald's sphere with respect to the lattice.

*Note: An alternative definition is that the size of Ewald's sphere
is changed with a change in wavelength. More specifically, some conventions
have the radius of Ewald's sphere as 1, with the reciprocal lattice scaled
by a multiplicative factor of the wavelength. The other convention has
the radius of Ewald's sphere as 1/l, where l is the wavelength with no
wavelength dependence on the size of the reciprocal lattice. We have chosen
the former so that the relative scaling of data at different wavelengths
is preserved for a given crystal to detector distance.

<<<Problem 4>>>

Problem 4

It is possible to use the simulator to integrate the diffraction over
some angular range around the Phi axis, resulting in a diffraction pattern
that is very much like an oscillation or rotation photograph or a single
frame in an area detector data set.

1. This is done by choosing the "Auto Rotation"
option to set parameters for the integration. The default parameters call
for a 3 degree rotation, with calculations of the spots every 0.2 degree.
These will be O.K. for this example. After you click on "OK",
then click on the "Start" box in
the lower left-hand corner. You can now see more of the reciprocal lattice
on the detector, with distinct rows and columns of spots.

The detector can be erased to start another exposure by clicking on
the "Clear" box. (The menu at the top of the screen will remain
disabled until you click "Clear".)

If one chooses large integration angles (or very large unit cells),
some spots overlap with previously recorded ones. This effect can be simulated
with this program, and is a real problem that restricts the amount of data
that can be collected in one integration.

<<<Problem 5>>>

Problem 5 (optional)

Space groups can be determined (with a few ambiguities) by examination
of the reciprocal lattice through X-ray diffraction patterns. By exploring
different crystal orientations with the slide bars, determine the likely
space groups of some "unknowns" whose parameters have been saved
in precreated files. The directions below describe how to load the unknowns.
(This is an advanced, difficult exercise, especially if done without peeking
at the reciprocal lattice!) If you want to check your answer you can open
the "Edit Unit Cell..." option and take a look at the cell constants
for the file you chosen as your unknown.

One important piece of information you need here is the geometrical
arrangement of the slide bar angles. They are arranged according to the
following diagram, which is like a standard goniometer head that is commonly
used in the laboratory.

1. Under the "File" option, select "Open", then
choose an unknown cell to attempt to determine. Several examples {/usr1/tutorials/}
are listed as "unknown1", "unknown2", "unknown3",
etc. . It will help if you understand the definitions of the the rotation
axes, Small, Large, and Phi (see below). Then you can plan your crystal
rotations to examine the diffraction from the crystal in different directions.

3. It will help to take some small, say 3.0 degree oscillation pictures
along different directions, as in Exercise 4.

4. If you need a clue, you can sneak a peek at the reciprocal lattice
itself by choosing the 'View Side" option.

Small corresponds to the small arc on a goniometer head, Large to the
large arc, and Phi is a rotation around the vertical axis. The arrows indicate
a positive rotation direction with all angles set to zero in this illustration.
Note that because Small is fastened above Large on the goniometer head,
as Large increases the Small axis of rotation changes. This is complicated,
but is the way real goniometer heads have to be constructed.

_
| |
|
|
|
\_______/ Small is a rotation perpendicular to Large.
|
\_________/ Large is a rotation in the plane of the screen.
|
___________ Phi is a rotation about the vertical axis
| |
|_________|

<<<Problem 6>>>

Problem 6

Synchrotrons can provide very intense X-ray beams with a wide range
of wavelengths. They can be useful for taking a data set very quickly,
sometimes in a millisecond or less. A so-called multi-wavelength "Laue
photograph" can be simulated by:

1. Setting the mosaic spread to the minimum
value (this reduces "streaking" of the spots on the detector,
and illustrates an actual limitation of the Laue method. It is also best
to choose unit cell constants in the 40 Angstrom range, with the highest
resolution set to 5 Angstroms.

2. The simulation is done by choosing the View Detector option (to save
computer time), turning Integration On (also
under the View menu header), and then choosing the Edit Wavelength box
and systematically decreasing the wavelength from 2.0 to 0.5 in 0.2 Angstrom
increments using the small right hand arrow and the pushing the Apply button.
(It's tedious, but it works).

Note that circles build up. The reflections at the high resolution limit
are no longer the farthest from the center of the pattern, but near the
intersections of circles, or nodal reflections.

Exercise 1

This exercise (Exercise1) is designed to familiarize you with the program
XtalView, and the process of building residues into an electron density
map. A model structure with a poly-alanine region is given in file Exercise1.pdb.
The experimental Structure Factors (Fo) obtained from diffraction data
are listed in file Exercise1.phs.
In this exercise reidues 90 -100 form a poly-alanine chain which you must
mutate to the correct sequence. The protein sequence for this crystal is
listed in Exercise1.seq.
Write down the sequence for residues 90-100, you will use this to create
the full model in Xfit.

The Xtalmanager window now appears. You will need to choose or create
a project and crystal entry before proceeding. ( The cell
dimensions and space group are listed in the file: /usr1/tutorials/exercise1/crystal_Ex1).

You may need to create a new directory for Exercise1 and copy the above
files from /usr1/tutorials/exercise1
into it. You cannot save your model into the /usr1/tutorials directory,
you only have permission to write into your own directories. See the UNIX
section below for help.

In a Menu Window:

This
is a command or new window button. [left mouse] click to execute

This
is a pull down menu. [right mouse] click and hold to see the menu

The
PIN in the upper left hand side of a pull down menu will 'stick' the window
open

After selecting a project and crystal file:

Choose
the application XFIT and list the files. Choose
the "*.pdb" and "*.phs" files for exercise1
by clicking on them.

Click (ADD ARGS) and (RUN).
This will bring up the XFIT program with the
Exercise1.pdb and Exercise1.phs files loaded.

XFIT

This first thing that should be done is to calculate the Structure Factors
and phases using the model (SFCalc) and from these generate the Electron
Density Map usinf a fast fourier transform (FFT).

To Mutate a residue go the the (MODEL)
window. Click on the residue you want to mutate, then choose the new residue
type that you want. When you have chosen the new residue type, pull down
the (Insert Res) menu and choose (Replace
and Fit). This fits the new residue into the electron density.

Rotate the model around and check how well the newly placed residue
fits the electron density. If the automatic fitting routine does not produce
the a good fit then a manual fitting of the residue may be required. The
tools for manually fitting a residue are in the pull down menus of the
(Xfit Tools) window. Under (Expert
Tools) are two other menus (Fit) and
(Middle Button Mode).

Fit ========
Chooses the number of residues/atoms from the stack to be fit, and also
applies your fit.

Exercise 2

Follow the instructions for xfit above, only
choosing project Exercise2.

In this exercise you will build a sequence of missing residues into
an electron density map. The Model, Exercise2.pdb,
is complete except for missing residues at 60 - 67. The sequence for these
missing residues is given in file Exercise2.seq
which will be read into xfit later.

The phase file Exercise2.phs contains
the Fobs, Fcalc, and model Phases, so that it can be used to generate an
electron density map without using the built-in Fcalc function (SFcalc).
So just use the (FFT) window to generate the
Electron density from the phase file. The 'Quadratic'
spline option produces smoother maps, but will be slower.

The baton command tool simplifies building of an a-carbon backbone into
electron density where no previous model exists. The baton commands can
be used either from the (Auto Fit)
pull down menu, or via the keyboard shortcuts shown.

The '>' command adds the next alpha-carbon
after the current residue. Once you have more than 5 CA's in a series (fragment)
then the 'Poly Ala Fragment' button can be
used to create a poly-alanine chain from the CA series. If this fails for
any residue then the individual CA (MRK) residue must be manualy mutated
to an alanine in the (MODEL) window and then
rotated into position using the (Middle Button)
commands. When done be sure to hit ';' to
exit Baton mode.

Go to the (FILES) window and read in the
sequence file /usr1/tutorials/exercise2/Exercise2.seqChange the directory to your home (/usr1/tutor01) and then ouput
(save) your model as model.pdb

In the (Baton) window choose 'Set
Sequence' to change the sequence of your poly-alanine fragment.

After the sequence is set it will need to be renumbered. In the (MODEL)
window choose a residue which has a correct reidue number and then pull
down {right_mouse_button} the (SEQUENCE)
menu and choose 'Renumber chain'.

Center on the first residue in the mutated fragment and choose (MODEL)(Insert
Res)'Replace and Fit'. Hit <space_bar>
to center on the next residue and repeat.

When you are finished with model building be sure to SAVE your
final model (FILE)'Output'.
It is good practise to save your working model at each step before major
changes are made. If you do not then your work may be lost.

The final model (model.pdb) is now ready to be refined using a more
sophisticated refinement program: XPLOR. (Make sure that the model.pdb
file does not contain any 'TER' entries and ends with an 'END')

and watch as the model parameters are refined in real-time. The final
output is a table of the cryatllographic R-factor during the refinement.
Notice how the R-factor decreases indicating an improvent in the model
as refinement proceeds.

Exercise 3

The structure determination of a new protein often starts from heavy
atom or Multiple Isomorphous Replacement (MIR) maps. The difference between
the diffraction from a native crystal and a crystal (or many crystals)
soaked in a heavy atom solution(s) is used to find the positions of the
heavy atoms in the unit cell. With several heavy atom derivatives combined
the 'PHASE PROBLEM' can be solved.

The Xtalmanager window now appears. You will need to choose or create
a project and crystal entry before proceeding. ( The cell dimensions and
space group are listed in the file: /usr1/tutorials/exercise3/crystal_Ex3
)

Choose
the program Xfft and list the files. Choose the Exercise3.df
derivative diffraction data file and the Exercise3.map
ouput file for exercise3. Note that the *.df
file has 4 columns two for each crystal (native and derivative). The two
columns per crystal are for anomalous measurements. If there is no anomalous
signal these two columns are identical.

Xcontour

Displays a countour map of the PATTERSON difference map generated in
Xfft. Remember that for space group 76 P41, the special planes are Z=0.0.
0.25, 0.50, 0.75. A heavy atom peak should be visible on the Z=0.25 plane.

This exercise (exercise5) will demonstrate the crystallographic refinement
of a point mutant using the software package XPLOR. The initial model will
be that of the native, unmutated, structure. The tyrasine at position 27
has been mutated to a tryptophan. Refinement and Modeling of a Point Mutant
In this example we have mutated one point residue Tyrosine 27 to a Tryptophan.

We will start this exercise with the following files: Exercise5.pdb
the native structure which is used as a starting model, Exercise5.cell
the experimental cell constants and symmetry elements and Exercise5.fob
the experimental diffraction intensities We will be using the refinement
program XPLOR to refine the atomic positions of all the atoms in the model
and the temperature (vibrational/disorder) factors associated with them.
The parameter files needed to perform these refinements are ready to be
used by XPLOR. The order in which these refinements should be performed
is given in the following files:

Please note the required names of all the input, and standard output
file names. Always save your model as you proceed, so that in the event
of an error you can recover your previous work. Record the 'R' factor after
each step of the refinement. Note how the "R" factor decreases
as the model is refined against the experimental data, and increases as
the resolution is increased.

use the unix command

grep "R=" *.log

to keep track of the R factor as it changes. Examine the model in XtalView's
XFIT program ( rigid.pdb and native.pdb)